Where Spec denotes the set of all prime ideals or R (or S).
V(I) is a subset of R that contains all the prime ideals that contain I.
Also, suppose a prime ideal P in S. f^(-1)(P) is a prime ideal in R.
Prove that f* is continuous.
I'm not sure if this is helpful. I proved that the preimage of a prime ideal in S is a prime ideal in R. Working off this knowledge, suppose a V(A) in R. Does this imply that there exists a p in P such that f^(-1)(p)=V(A)?